Theory of local orbital magnetization: local Berry curvature

Fr\'ed\'eric Pi\'echon; Karyn Le Hur; Sariah Al Saati

REVIEW 2 major objections 5 minor 3 cited by

Theory of local orbital magnetization: local Berry curvature

T0 review · 2 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper derives a single-site formula for orbital magnetization in tight-binding systems, decomposing it into a local orbital moment and a local Berry curvature that together capture topology in finite samples.

desk verdict A useful local orbital magnetization marker, but the k-space derivation rests on a false operator identity in the supplement. read the letter →

arxiv 2512.12343 v2 pith:YBQAQLUC submitted 2025-12-13 cond-mat.mes-hall

Sariah Al Saati , Karyn Le Hur , Frédéric Piéchon This is my paper

classification cond-mat.mes-hall

keywords localorbitalmagnetizationBerrycurvaturetight-bindingmodelssublatticetexturetopologicalinsulatorChernmarkerfinite-sizesystemsHaldanemodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper derives a microscopic, thermodynamic formula for the orbital magnetization at a single atomic site in a tight-binding system. Each occupied eigenstate contributes two local pieces: the conventional orbital magnetic moment weighted by the on-site probability, and a newly identified local Berry curvature that redistributes electronic weight under a magnetic field. The same formula is obtained in Bloch (k-space), open-boundary (r-space), and ribbon geometries, and it reduces to the usual total orbital magnetization when summed over all sites. Applied to two-band honeycomb models, it exposes orbital ferromagnetic order in a topological insulator and orbital antiferro- and ferrimagnetic textures in trivial insulators. The authors argue this resolves topology locally in the bulk of finite systems without invoking edge states.

What carries the argument

The engine is the gauge-invariant perturbative expansion of the magnetic-field-dependent local density of states: Morb(r) is defined as −∂_B Ξ(r,μ,B)|_{B=0}, with the first-order correction expressed through the zero-field Green's function G, velocity operator v̂ = −i[r̂,H], and the cross product G(v̂G)×(v̂G). Spectral decomposition of that expression yields the two-term form m_n(r)+Ω_n(r); in k-space the local Berry curvature separates into Ω^topo (which sums to the usual Berry curvature) and Ω^geom (which sums to zero). This decomposition does the work: it is what makes topological information visible at a single site.

What would settle it

Take a small open-boundary tight-binding cluster and compute Morb(r) using the paper's Green's-function expression; independently evaluate −[Ξ(r,μ,B)−Ξ(r,μ,−B)]/(2B) by exact diagonalization of the Peierls-substituted Hamiltonian and extrapolate to B→0. If the two disagree at any bulk site, or if the result changes under a gauge transformation of the uniform magnetic field, the identification of the local magnetization with this derivative fails.

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Extended reading notes

Core claim

The central claim is that the local orbital magnetization Morb(r) at site r equals (e/ℏ)Σ_n [f(ε_n) m_n(r) + F(ε_n) Ω_n(r)], where m_n(r) is the eigenstate's orbital magnetic moment projected on site r and Ω_n(r) is an effective local Berry curvature built from interband matrix elements of r̂×v̂. In the Bloch representation, the sublattice Berry curvature splits into a topological part that sums to the usual Berry curvature and a geometric part that sums to zero. Numerical tests on two-band honeycomb models show the k-space, r-space, and ribbon computations agree at bulk sites, while an earlier local Chern marker gives different site-resolved values. The paper concludes that the local Berry

Load-bearing premise

That the zero-field B-derivative of the local grand-canonical potential is the physically observable single-site orbital magnetization, and that the first-order perturbative expansion in the Peierls phase captures a gauge-invariant local response.

Editorial extensions

If this is right

Orbital magnetization can be mapped at sublattice resolution in crystals, revealing magnetic orders invisible in the total magnetization.
In finite open-boundary samples, the local Berry curvature of bulk band states accounts for the quantized slope of magnetization in the topological gap, so topology can be read locally without edge states.
The same formulas apply to any tight-binding model, including amorphous, quasicrystalline, moiré, and molecular systems where Bloch momentum is not available.
The site-resolved computation scales as N², against N³ for the earlier local Chern-marker expression, making large finite samples tractable.
The new local Chern marker C(r) = 2π Σ_n f(ε_n)Ω_n(r) obeys the same sum rules as the existing marker but gives different sublattice values, so it is a distinct local topological probe.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the authors leave implicit: integrating the local Berry curvature over a region around a defect or dislocation should recover the topological charge carried by that region, giving a fully local bulk-edge correspondence.
The geometric contribution Ω^geom, which vanishes only after summing over sublattices, suggests that site-resolved orbital magnetization textures could serve as a real-space diagnostic for band-geometric quantities beyond Berry curvature, such as the quantum metric in two-band systems.
A testable experimental consequence is that the predicted sublattice ferro- or antiferromagnetic orbital order should appear as persistent alternating current-loop patterns in the bulk of patterned topological samples, observable with local magnetometry at zero applied field.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper has a genuinely useful idea: a single-site decomposition of orbital magnetization, plus a local Berry curvature marker that works in finite and aperiodic systems at lower cost than Bianco-Resta. The r-space formulas (8)–(10) are derived cleanly from the Green's function expression (56), and the numerical agreement across k-space, r-space, and ribbon geometries is strong evidence that the local quantities are well-defined and useful. The comparison with Bianco-Resta is also fair and informative, and the N^2 cost advantage is real.

The soft spot is concrete and load-bearing for the k-space half of the paper. The supplement derives the starting equation (2) from equation (56) using the identity G r̂ = G v̂ G (line after Eq. (56), referencing Eq. (68)). That identity is false. With v̂ = -i[r̂,H], Eq. (68) gives G v̂ G = -i[r̂,G] = i(G r̂ - r̂ G), hence G r̂ = r̂ G - i G v̂ G, not G r̂ = G v̂ G. A two-site chain already contradicts it. Since Section 3.2 of the supplement begins from equation (57), which is obtained via this false identity, the sublattice magnetization and the topological/geometric split of the Berry curvature are not established as written. The numerical agreement with the r-space results suggests the final k-space formulas might be correct, but a match with an independent calculation does not repair the proof gap.

The paper cites Raoux et al. for Eq. (2), so it is possible the equation is correct even though the supplement's derivation of it is wrong. That needs to be checked and, if correct, derived properly. As it stands, a serious referee should ask the authors to fix this before the k-space claims can be trusted. The r-space part and the numerical results are solid enough to justify that effort.

Overall: worth reviewing, not acceptable in current form. The authors have found an appealing tool and done honest numerical work, but they need to supply a valid derivation for the k-space decomposition. I would not cite it yet.

Referee Report

2 major / 5 minor

Summary. The manuscript develops a thermodynamic theory of local, single-site orbital magnetization in tight-binding models. Starting from a Green's-function expression for the derivative of the local grand potential with respect to a magnetic field, it derives sublattice-resolved Bloch expressions for the orbital magnetization, decomposes the resulting local Berry curvature into topological and geometric parts, and derives real-space expressions for open-boundary systems and ribbon geometries. The paper reports numerical agreement across k-space, r-space, and ribbon calculations for the Haldane model and a modified Haldane model, identifies sublattice orbital ferro-, antiferro-, and ferrimagnetic textures, and proposes a local Berry-curvature marker as a bulk probe of topology in finite systems.

Significance. If the central derivation is sound, the paper offers a genuinely useful microscopic tool: explicit, parameter-free single-site formulas for local orbital magnetization and local Berry curvature that apply equally to periodic, open, and ribbon geometries and, in principle, to aperiodic systems. The supplement is valuable: it contains detailed analytic derivations, and the numerical cross-checks across three geometries and three models are a strength. The proposed decomposition into topological and geometric sublattice Berry curvatures is conceptually significant and goes beyond the Bianco-Resta local marker. The main weakness is that the derivation of the k-space formulas rests on a false operator identity; this must be repaired or explicitly outsourced to Ref. [9] before the k-space results can be regarded as established.

major comments (2)

[SM Sec. 2, Eqs. (56)-(57), (68)] The passage from Eq. (56) to Eq. (57) uses the stated identity G r = G v G. This identity is false. With v = -i[r,H], Eq. (68) actually gives G v G = -i[r,G] = i(G r - r G), hence G r = rG - i G v G. A concrete counterexample is the two-site chain H = [[0,t],[t,0]], r = diag(0,a), for which G r and G v G differ. Since Sec. 3.2 starts from Eq. (57), the k-space sublattice decomposition, Eqs. (3)-(7) and SM Eqs. (99)-(102), and the topological/geometric split are not proven by the supplied derivation. The r-space and ribbon derivations start from Eq. (56) and remain valid, but numerical agreement with the k-space results cannot repair the missing proof. The authors should either quote Eq. (2) directly from Ref. [9] or provide a correct derivation of Eq. (57) from Eq. (56).
[Main text Eq. (2) and SM Sec. 2] The identification Morb(r) = -d_B Xi(r,mu,B)|_{B=0} is assumed rather than derived, and the gauge/origin independence of this single-site quantity is not discussed. This may be acceptable as an import from Ref. [9], but the manuscript should state that explicitly; as written, the local Chern marker C(r) and all sublattice texture results rest on this identification. A short demonstration that the derivative of the local grand potential is the appropriate physical local observable would remove the ambiguity.

minor comments (5)

[Figs. 1-2] Typo in labels: 'Mofified' should be 'Modified'; similarly, Fig. 3 caption has 'midle' for 'middle'.
[Eqs. (4)-(7)] The summation results are marked 'in red,' which is lost in grayscale or monochrome printing. Please state these results explicitly in the text.
[SM Sec. 2 around Eq. (56)] The notation v = H r is confusing; if H is the Hamiltonian, H r is a product, not a commutator. Use v = -i[r,H] consistently throughout.
[Main text Eq. (11)] At T=0 the relation F(epsilon) = (mu-epsilon) f(epsilon) is exact in the distributional sense, so the approximate symbol 'simeq' can be replaced by an equality.
[Fig. 3] The separation into 'band states' and 'gap states' in a finite-size spectrum should be defined precisely (e.g., by an energy window), since the claim about where topology is encoded depends on this distinction.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new local decompositions are derived algebraically from an externally established starting point; no fitted parameter or prediction reduces to the input.

full rationale

The paper's central results, Eqs. (3)-(7) and (8)-(10), are algebraic consequences of Eq. (2), which is explicitly imported from Raoux et al. (ref. [9]) and reviewed in the supplementary material. That citation is a self-citation (F. Piéchon is a co-author of both works), but it is disclosed and Eq. (2) is a published, parameter-free input whose assumptions do not include the local Berry curvature decomposition claimed here. The new contributions are not fitted to any data, nor are they renamed outputs: the k-space and r-space derivations are done by spectral decomposition, partial-fraction integration, and summation identities, and the 'topological'/'geometric' split is generated by which terms survive sublattice summation. The numerical comparisons among k-space, r-space, and ribbon geometries are cross-checks of the same derived objects, not fits or definitions. There is no uniqueness theorem, no ansatz smuggled in by citation, and no parameter renamed as a prediction. The only flagged concern in the review is the supplementary statement 'using the identity G r̂ = G v̂G (see equation 68 below)'; if that identity is in error, it would be a mathematical-validity problem in the derivation of Eq. (2), not a circularity, because it does not make the claimed prediction equivalent to an input. Accordingly, no circularity step can be exhibited, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central formulas are exact identities for non-interacting tight-binding models; the only external inputs are the Peierls-substitution perturbation theory of Raoux et al. and the model Hamiltonians used in the numerical illustrations. The numerical examples use specific parameter choices (t2=1/3, M=√3/10 etc.) but these are not fitted; they are selected to realize topological/trivial phases.

assumptions (4)

domain assumption The system is described by a non-interacting spinless tight-binding Hamiltonian.
Stated in the text after Eq. (2): 'considering non-interacting (spinless) electrons in a tight-binding picture'.
domain assumption The coupling to a uniform magnetic field is given by Peierls substitution, and first-order perturbation theory in B is valid.
SM Section 2: G(B)=G+B·∂B G|_0, using Peierls phases; valid for weak fields.
domain assumption The local orbital magnetization is defined as M(r,μ,B)=−∂_B Ξ(r,μ,B)|_{B→0} with the local grand potential from the field-dependent local density of states.
Main text Eq. (2) and SM Eq. (23); this definition is taken from Raoux et al. [9].
domain assumption The gauge-invariant perturbative expression for the first-order field-induced correction to the Green's function (SM Eq. 54) is correct.
SM Section 2 reviews the Raoux et al. derivation; this is the starting point for all subsequent formulas.

invented entities (1)

Local Berry curvature Ω_n(r) (and its sublattice version Ω_nk(rα))
purpose: Serves as a single-site real-space marker that redistributes orbital magnetization and encodes topology in finite/aperiodic systems.
Defined from the velocity and position operators in Eqs. (10) and (5)–(7); it is an exact derived quantity, not a postulated physical object. It has no direct experimental observable independent of the paper; its value is assessed through internal consistency (k/r agreement) and recovery of the Chern number in examples.

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Pith. "Pith review of Theory of local orbital magnetization: local Berry curvature." pith.science (2026). https://pith.science/paper/YBQAQLUC

@misc{pith2026251212343,
  author       = {Pith},
  title        = {Pith review of: Theory of local orbital magnetization: local Berry curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBQAQLUC}},
  note         = {Machine review of arXiv:2512.12343}
}

read the original abstract

We develop a thermodynamic theory of local orbital magnetization based on a perturbative expansion of the magnetic-field-dependent local density of states. Applicable to periodic crystals, finite systems with open boundaries, and ribbons alike, the formalism resolves orbital magnetic textures at the sublattice scale. It further reveals a previously unidentified local Berry curvature that captures the magnetic-field-induced redistribution of electronic weight in real space. We establish the consistency of the theory across geometries, identify orbital ferro-, antiferro-, and ferrimagnetic phases in both topological and trivial insulators, and demonstrate that the local Berry curvature provides a bulk description of topology in finite systems.

Figures

Figures reproduced from arXiv: 2512.12343 by the authors.

**Figure 1.** Sublattice orbital magnetizations MA (red) and MB (blue); MA + MB (black). In gray, we highlight the gap region. We observe in the gap that the slope of the black curve is quantized and satisfies exactly ∂µ(MA + MB) = e hC, even locally in r-space. Rows: (top) HMtopo, (middle) HMtriv,(bottom) mHM. Columns: (Left) results for Bloch electron using (4,5,6,7), (Right) results obtained using (9,10), computed locally on a… view at source ↗

**Figure 2.** Magnetization for a single site (chosen arbitrar [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Orbital magnetization in the topological phase [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 1.** Figure 1: (Left) Honeycomb lattice described as a lattice of two non-equi [PITH_FULL_IMAGE:figures/full_fig_p007_1.png]

**Figure 2.** Figure 2: Next-nearest neighbor hopping term for the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png]

**Figure 3.** Figure 3: Total orbital magnetization in the three geometries. Each panel sh [PITH_FULL_IMAGE:figures/full_fig_p020_3.png]

**Figure 4.** Figure 4: Local orbital magnetization in the three geometries. The blue and [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

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Forward citations

Cited by 3 Pith papers

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Reference graph

Works this paper leans on

7 extracted references · cited by 3 Pith papers

[1]

F. D. M. Haldane. Model for a Quantum Hall Eﬀect without Landau Levels: Con densed-Matter Realization of the ”Parity Anomaly”. Physical Review Letters, 61(18):2015–2018, October 1988. ISSN 0031-9007. doi: 10.1103/PhysRevLett.61.2015. 17

doi:10.1103/physrevlett.61.2015 2015
[2]

Orbital magnetism in coupled-bands models

Arnaud Raoux, Fr´ ed´ eric Pi´ echon, Jean-No¨ el Fuchs, and Gilles Montambaux. Orbital magnetism in coupled-bands models. Physical Review B , 91(8), February 2015. ISSN 1550-235X. doi: 10.1103/physrevb.91.085120

doi:10.1103/physrevb.91.085120 2015
[3]

R. Peierls. Zur theorie des diamagnetismus von leitungselektron en. Zeitschrift fur Physik , 80 (11–12):763–791, November 1933. ISSN 1434-601X. doi: 10.1007/bf01342591

doi:10.1007/bf01342591 1933
[4]

Thonhauser, Davide Ceresoli, David Vanderbilt, and R

T. Thonhauser, Davide Ceresoli, David Vanderbilt, and R. Resta. Orbital magnetization in periodic insulators. Physical Review Letters , 95(13), September 2005. ISSN 1079-7114. doi: 10.1103/physrevlett.95.137205

doi:10.1103/physrevlett.95.137205 2005
[5]

Thonhauser, David Vanderbilt, and R

Davide Ceresoli, T. Thonhauser, David Vanderbilt, and R. Resta. Orbital magnetization in crystalline solids: Multi-band insulators, chern insulators, and m etals. Physical Review B , 74 (2), July 2006. ISSN 1550-235X. doi: 10.1103/physrevb.74.024408

doi:10.1103/physrevb.74.024408 2006
[6]

Thonhauser

T. Thonhauser. Theory of orbital magnetization in solids. International Journal of Modern Physics B , 25(11):1429–1458, April 2011. ISSN 1793-6578. doi: 10.1142/s0217979211058912

doi:10.1142/s0217979211058912 2011
[7]

Chern number and orbital magnetization in ribbons, polymers, and single-layer materials

Enrico Drigo and Raﬀaele Resta. Chern number and orbital magnetization in ribbons, polymers, and single-layer materials. Physical Review B , 101(16), April 2020. ISSN 2469-9969. doi: 10.1103/physrevb.101.165120. 18

doi:10.1103/physrevb.101.165120 2020

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

F. D. M. Haldane. Model for a Quantum Hall Eﬀect without Landau Levels: Con densed-Matter Realization of the ”Parity Anomaly”. Physical Review Letters, 61(18):2015–2018, October 1988. ISSN 0031-9007. doi: 10.1103/PhysRevLett.61.2015. 17

doi:10.1103/physrevlett.61.2015 2015

[2] [2]

Orbital magnetism in coupled-bands models

Arnaud Raoux, Fr´ ed´ eric Pi´ echon, Jean-No¨ el Fuchs, and Gilles Montambaux. Orbital magnetism in coupled-bands models. Physical Review B , 91(8), February 2015. ISSN 1550-235X. doi: 10.1103/physrevb.91.085120

doi:10.1103/physrevb.91.085120 2015

[3] [3]

R. Peierls. Zur theorie des diamagnetismus von leitungselektron en. Zeitschrift fur Physik , 80 (11–12):763–791, November 1933. ISSN 1434-601X. doi: 10.1007/bf01342591

doi:10.1007/bf01342591 1933

[4] [4]

Thonhauser, Davide Ceresoli, David Vanderbilt, and R

T. Thonhauser, Davide Ceresoli, David Vanderbilt, and R. Resta. Orbital magnetization in periodic insulators. Physical Review Letters , 95(13), September 2005. ISSN 1079-7114. doi: 10.1103/physrevlett.95.137205

doi:10.1103/physrevlett.95.137205 2005

[5] [5]

Thonhauser, David Vanderbilt, and R

Davide Ceresoli, T. Thonhauser, David Vanderbilt, and R. Resta. Orbital magnetization in crystalline solids: Multi-band insulators, chern insulators, and m etals. Physical Review B , 74 (2), July 2006. ISSN 1550-235X. doi: 10.1103/physrevb.74.024408

doi:10.1103/physrevb.74.024408 2006

[6] [6]

Thonhauser

T. Thonhauser. Theory of orbital magnetization in solids. International Journal of Modern Physics B , 25(11):1429–1458, April 2011. ISSN 1793-6578. doi: 10.1142/s0217979211058912

doi:10.1142/s0217979211058912 2011

[7] [7]

Chern number and orbital magnetization in ribbons, polymers, and single-layer materials

Enrico Drigo and Raﬀaele Resta. Chern number and orbital magnetization in ribbons, polymers, and single-layer materials. Physical Review B , 101(16), April 2020. ISSN 2469-9969. doi: 10.1103/physrevb.101.165120. 18

doi:10.1103/physrevb.101.165120 2020